How does mass create curvature in spacetime?

[Passionflower]

While clocks in an accelerating rocket show Doppler shift with height as in a gravitational field, sequentially dropped objects in the rocket will maintain constant mutual spatial separation, demonstrating that the presence or absence of intrinsic curvature of the background can be detected even in an accelerating rocket.

I am not sure I understand this.

In a linearly accelerating rocket if we drop one object at v(t) and another one at v(t+1) then the dropped objects will have a relative velocity wrt each other right? In other words their separation will grow over time.

 

[yuiop]

I am not sure I understand this.

In a linearly accelerating rocket if we drop one object at v(t) and another one at v(t+1) then the dropped objects will have a relative velocity wrt each other right? In other words their separation will grow over time.

Oops, your right. One will have constant velocity relative to the other. I wonder if the distinction is that in a real gravitational field with intrinsic curvature, one will be accelerating relative to the other?

 

[pervect]

Gravitatonal time dilation is a REALLY BIG clue that space-time is curved. If you think of gravity as just being a force, there's no reason it should affect how clocks operate. But we observe that it does. Recent experiments can detect the gravitational time dilation for a height difference as small as a foot.

When we draw the resulting space-time diagram, we have diagrams that look like they should be parallelograms, but the sides are of unequal length. (The "sides" of unequal length in this diagram are the clocks which run at different length because it's a space-time diagram).

This isn't possible in an Eulidean geometry, and suggests (as most textbooks will mention) that space-time is curved.

To actually calculate a curvature invariant and show that it is nonzero is a bit more involved, but it's certainly possible in principle. It involves measuring the metric accurately enough to get the second partial derivatives. This will rule out some special cases such as being in an elevator. But basically, the reason we know that space-time is curved is that clocks appear to run at different rates depending on their gravitational potential.

 

[yuiop]

Gravitatonal time dilation is a REALLY BIG clue that space-time is curved. If you think of gravity as just being a force, there's no reason it should affect how clocks operate. But we observe that it does. Recent experiments can detect the gravitational time dilation for a height difference as small as a foot.

Isn't there a problem with this, in that clocks lower down in an accelerating rocket will tick slower and red shift relative to clocks higher up in the rocket even in the rocket is accelerating in flat space? In other words, differential clock rates due differences in "height" does not seem in itself to be proof of intrinsic spacetime curvature.

 

[Passionflower]

Isn't there a problem with this, in that clocks lower down in an accelerating rocket will tick slower and red shift relative to clocks higher up in the rocket even in the rocket is accelerating in flat space? In other words, differential clock rates due differences in "height" does not seem in itself to be proof of intrinsic spacetime curvature.

We could claim that the clocks run at a different rate because of the difference in curvature (that is path curvature, not spacetime curvature) of the worldlines, just like in the case of an accelerating rocket.

However in curved spacetime there is of course an additional factor.

Consider the moment when a free falling observer and a stationary observer are both at location r=4 in a Schwarzschild solution where m=1/2, then:

The stationary observer measures a distance to the EH of

<br /> \sqrt {r \left( r-1 \right) }+\ln \left( \sqrt {r}+\sqrt {r-1}<br /> \right) = 4.781059513<br />

The free falling observer measures a distance of 3 while the Fermi normal distance is: 3.488654305.

The local velocity between the stationary and free falling observer is:

<br /> \sqrt {{r}^{-1}} = 0.5<br />

So if we (of course falsely) assume for a moment that spacetime is flat we would expect the distance to the EH for the free falling observer from the perspective of the stationary observer to be simply the Lorentz contracted distance. However this is not the case, as shown as follows:

The Lorentz factor becomes:

<br /> {\frac {1}{\sqrt {1-{v}^{2}}}} = 1.154700538<br />

And thus the (assuming flat spacetime) distance would be:

4.781059513 / 1.154700538 = 4.140518997

Which is different from 3 (or a Fermi normal distance of: 3.488654305) as measured by the free falling observer.

And from the perspective of the free falling observer when applying the reverse Lorentz factor he calculates a distance of 3.464101617 (or 4.028351006 from the perspective of a Fermi normal distance) as opposed to 4.781059513.

So in a Schwarzschild spacetime we see a kind of 'bifurcation' of spatial distances. This by the way also demonstrates that Lorentz covariance applies only locally in GR.

Here is a demonstration of this, the graphs are a little complicated, notice that the Y axis is Distance/Schwarzschild Coordinate -1:

[PLAIN]http://img258.imageshack.us/img258/7867/001bifurcation2.gif [PLAIN]http://img245.imageshack.us/img245/4461/001bifurcation1.gif

 

[PAllen]

I wonder if the following scenario clarifies any of these interpretational questions.

Suppose we compare our universe to one in which gravity only affects matter with either color or e/m charge (in particular, neither photons nor neutrinos are affected by gravity). In such a universe, you could unambiguously distinguish measurements of gravity from measurement of what background geometry there is for spacetime.

Comparing that universe to ours, you can say with equal justification, that gravity is no longer a force, instead mass/energy interacts with geometry as specified by GR; or that geometry has become unobservable because it is indistinguishable from gravity as determined by GR. If this is what Mentz114 is getting at, then I suspect few would have any disagreement. This distinction is purely philosophic, and there is no objective criterion you can use to say one is a more economic viewpoint than the other. Purely a matter of taste.

 

[Dale]

Oops, your right. One will have constant velocity relative to the other. I wonder if the distinction is that in a real gravitational field with intrinsic curvature, one will be accelerating relative to the other?

Yes, this is correct. In a flat spacetime two inertial objects may have a relative velocity, but that relative velocity will be constant. In a curved spacetime two inertial objects may not only have a relative velocity, but that relative velocity may change.

 

[Naty1]

I think Mentz comment reflects my own view:

I must disagree, because you're not measuring curvature. You're fitting observations to a theory that uses curvature to model the effects of gravity.

and also this from pervect:

Gravitatonal time dilation is a REALLY BIG clue that space-time is curved.

but this goes perhaps too far:

...the reason we know that space-time is curved is that clocks appear to run at different rates depending on their gravitational potential.

By coincidence I opened Kip Thorne's BLACK HOLES AND TIME WARPS for a read outside in the sun this morning.. (This can likely provide fodder for different views expressed above!)

...In deducing his equivalence principle, Einstein ignored tidal gravitational forces...by imagining that you and your reference frame are very small...from mid 1911 to mid 1912 Einstein tried to explain tidal gravity by assuming that time is warped but space is flat...he abandoned this approach in favor of warped space and time...because warpage 'smelled right' to Einstein...Einstein was a professor in Prague in 1912 when he realized tidal gravity and spacetime curvature are one and the same thing...

[Thorne says he got all this from reading the Russian translation of Einstein's original work...he does not read German. ]

Thorne goes on to say Einstein believed the law of warpage should look the same in every reference frame...but he and his professor buddy, Grossman, could not initially find equations which accomplished that...so they published in late 1912 their first attempt at general relativity...but it wasn't quite right...and a week before he finally arrived at the equations we know today in 1915 David Hilbert actually arrived at the correct formulation after Einstein discussed all his ideas with Hilbert ...not via Einsteins trial and error but with an "elegant, succinct mathematical route.."

So it seems clear to me these guys did what Mentz suggested above... "fitting observations to theory"...or another way to say the same thing, they described effects rather than causes...which goes to address the OP's initial question...

So Newton explained tidal forces by a difference in the direction and strength of gravity,
Einstein selected curved spacetime.

just very clever guesswork?? you decide...

 

[PAllen]

I think Mentz comment reflects my own view:
and also this from pervect:

but this goes perhaps too far:

By coincidence I opened Kip Thorne's BLACK HOLES AND TIME WARPS fo a read outside in the sun this morning.. (This can likely provide fodder for different views expressed above!)



[Thorne says he got all this from reading the Russian translation of Einstein's original work...he does not read German. ]

Thorne goes on to say Einstein believed the law of warpage should look the same in every reference frame...but he and his professor buddy, Grossman, could not initially find equations which accomplished that...so they first published in late 1912 their first attempt at general relativity...but it wasn't quite right...and a week before he finally arrived at the equations we know today in 1915 David Hilbert actually arrived at the correct formulatiion after Einstein discussed all his ideas with Hilbert ...not via Einsteins trial and error but with an "elegant, succinct mathematical route.."

So it seems clear to me these guys did what Mentz suggested above... "fitting observations to theory"...or another way to say the same thing, they described effects rather than causes...which goes to address the OP's initial question...

So Newton explained tidal forces by a difference in the direction and strength of gravity,
Einstein selected curved spacetime.

just very clever guesswork?? you decide...

I would describe it as inspired guesswork along the following lines:

Einstein wanted one concept, inertia, to completely replace two concepts, inertia and gravity. For tidal gravity, the problem is that inertial paths can converge and cross, and relative speeds can change between them. Combined with a view that inertia is the tendency to follow a 'straight line' when there is no applied force, it's hard to propose any alternative to what Einstein chose. Maybe the more inspired idea was, despite the issue of tidal gravity, Einstein persisted with the goal of having only one concept, inertia.

 

[Mentz114]

I wonder if the following scenario clarifies any of these interpretational questions.

Suppose we compare our universe to one in which gravity only affects matter with either color or e/m charge (in particular, neither photons nor neutrinos are affected by gravity). In such a universe, you could unambiguously distinguish measurements of gravity from measurement of what background geometry there is for spacetime.

Comparing that universe to ours, you can say with equal justification, that gravity is no longer a force, instead mass/energy interacts with geometry as specified by GR; or that geometry has become unobservable because it is indistinguishable from gravity as determined by GR. If this is what Mentz114 is getting at, then I suspect few would have any disagreement. This distinction is purely philosophic, and there is no objective criterion you can use to say one is a more economic viewpoint than the other. Purely a matter of taste.

That is well put and looks very plausible.

But this "... mass/energy interacts with geometry ..." can only be meaningful if the 'geometry' has some physical existence. It has no mass, charge, smell, colour, entropy, temperature nor 'hair' of any kind. So how can it react with anything ?

G(T)R is a recipe for working out geodesics, principally, and we can compare actual trajectories ( measured with clocks and rulers) to those predictions. If they fit we conclude that GR is a good model. It is tempting to assume then that the internal variables of the theory must actually be present, but the theory neither requires it nor provides any evidence for that assumption.

This is why there is a school of thought that asserts that gravity is geometry(curvature) and there is no causal path, or interaction required.

 

[Dale]

But this "... mass/energy interacts with geometry ..." can only be meaningful if the 'geometry' has some physical existence. It has no mass, charge, smell, colour, entropy, temperature nor 'hair' of any kind. So how can it react with anything ?

Why would you think that geometry requires any "hair" to interact with mass/energy? For example, a photon does not require charge in order to be able to interact with charged particles. The purely geometric properties such as distance, time, angles, velocities, and curvature seem plenty "hairy" to me.

G(T)R is a recipe for working out geodesics, principally, and we can compare actual trajectories ( measured with clocks and rulers) to those predictions. If they fit we conclude that GR is a good model. It is tempting to assume then that the internal variables of the theory must actually be present, but the theory neither requires it nor provides any evidence for that assumption.

Again, how is that different from ANY measurement? See posts 27 and 28. You seem to think that curvature is a fundamentally different thing than other measurements in some way that I don't understand.

 

[Mentz114]

Why would you think that geometry requires any "hair" to interact with mass/energy? For example, a photon does not require charge in order to be able to interact with charged particles. The purely geometric properties such as distance, time, angles, velocities, and curvature seem plenty "hairy" to me.

Geometry- vectors, lines angles and the stuff of geometry are abstractions that we carry in our minds. I ask you again, what physical embodiment of geometry is it that interacts with the sources of gravity ? If you believe that 'spacetime' is stuff, that's fine by me, maybe it is.

I believe in geometry, I like geometry. But it is mathematics. I must believe that there is an objective universe, but our attempts to understand it with our 'geometry' fall far short of capturing true complexity of the cosmos, which I also believe is beyond our understanding.

The general theory of relativity does not demand or require that its internal quantities should have physical correlates, only that the predictions, expressed in terms of length, time, mass and charge should agree with observations.

Again, how is that different from ANY measurement? See posts 27 and 28. You seem to think that curvature is a fundamentally different thing than other measurements in some way that I don't understand.

Let's try this, you think that measurements of curvature and force are just "fitting observations to a theory" and I assume that you think that other quantities are somehow "more"* than this. So what measured quantities would you place in the "fitting observations to a theory", what quantities would you place in the "more" category, and what do you think distinguishes measurements of the two?

Lets talk about Bob, who is performing these activities.
Measurement is the act of interaction which results in a single (scalar) quantity. In QM language, a system interacts with a measuring device causing the collapse of a wave-function, and an observable assuming a definite state.
I repeat - Bob is not measuring spacetime curvature - it is impossible. He is collecting data ( mass, length, time, charge) which will be used to make an estimate of an abstract multivalued parameter of a model or theory. This will be used to calculate trajectories, which will be compared to data to see if the model fits. At no time was 'curvature' being measured. We can't even measure a vector in one measurement, never mind a rank-4 tensor. If you use a formula to calculate something, it is not a direct measurement and you have to ask - where did the formula come from ?I hope we can agree to disagree, because I see no profit in carrying on with what seems an undecidable, until more data is available. I don't think either of us can produce an argument that can settle this.

For me it just comes down to this - if spacetime curvature has physical existence then so must spacetime, and there's no evidence for that at present.

 

[Dale]

Geometry- vectors, lines angles and the stuff of geometry are abstractions that we carry in our minds.

I strongly disagree with this. The fact that my table's leg is perpendicular to the table top is not merely a mental abstraction. The word "perpendicular" may be a mental abstraction, but the geometric relationship between the top and the leg is not.

Measurement is the act of interaction which results in a single (scalar) quantity. ... We can't even measure a vector in one measurement, never mind a rank-4 tensor.

Yes, we can agree to disagree. According to you only scalars qualify as "measured" quantities and all vector and tensor quantities are data "fit to a model". By your categorization I agree that curvature would be classified as a "fit to a model" since it is a tensor and other quantities would be "more" since they are scalars. I would not use your definition, but since it is clear and applicable any further argument on that point would be purely semantic. I am perfectly content to have curvature in the same category as momentum, position, and other such "fit to model" quantities.

 

[Naty1]

Mentz posts:

But this "... mass/energy interacts with geometry ..." can only be meaningful if the 'geometry' has some physical existence. It has no mass, charge, smell, colour, entropy, temperature nor 'hair' of any kind. So how can it react with anything ?

Gasp! Oh, the pooh bahs of the forum will reprimand and chastise you for such thoughts! I still remember posting something almost identical several years ago and the list of howls and complaints that followed was incredible...even after I subsequently listed 8 or 10 or more examples...THAT was when I first realized the orthodoxy that prevails here...

Dalespam posts:

Why would you think that geometry requires any "hair" to interact with mass/energy? For example, a photon does not require charge in order to be able to interact with charged particles. The purely geometric properties such as distance, time, angles, velocities, and curvature seem plenty "hairy" to me.

The first two sentences seem rather contradictory...The photon- electron interaction is energy based, like spacetime warpage, BUT can't we can more directly observe some resultant effects ...absorption, energy band change, deflections, emissions, and so forth?? c...not so with spacetime warpage...

Now the second part IS really interesting...I'm not sure I ever thought about whether such abstract characteristics have a physical interpretation when there are no underlying physical observables...

and:

Geometry- vectors, lines angles and the stuff of geometry are abstractions that we carry in our minds.

I strongly disagree with this. The fact that my table's leg is perpendicular to the table top is not merely a mental abstraction. The word "perpendicular" may be a mental abstraction, but the geometric relationship between the top and the leg is not.

But when neither the table nor the leg is an observable, as spacetime, such an angle sure seems "abstract" to me...

 

[Mentz114]

Yes, we can agree to disagree.

Thanks.

[ Sorry about the use of capitals in my post - I've edited it out and also corrected a spelling error]

 

[Dale]

But when neither the table nor the leg is an observable, as spacetime, such an angle sure seems "abstract" to me...

This seems to be a comment that deliberately ignores much of the current discussion. I gave a bunch of concrete observable phenomena above. We are not talking about mere abstract geometry here. We are talking about the geometrical relationships between physical events, matter, energy, waves, objects, etc. In concrete and measurable ways these geometrical relationships between physical events deviate from flatness. That is what is meant by saying that spacetime is curved.

Do you agree that a table leg may be orthogonal (a geometrical relationship) to a table top in a physical and non-abstract sense? What about waves, worldlines of particles, events on an object's worldline, etc.? Where do you draw the line between abstract and concrete in terms of geometrical relationships between physical events?

 

[Naty1]

Dalespam...

This seems to be a comment that deliberately ignores much of the current discussion. I gave a bunch of concrete observable phenomena above. We are not talking about mere abstract geometry here. ...Where do you draw the line between abstract and concrete in terms of geometrical relationships between physical events?

The question is what the observables measure...the issue I think is in your last sentence..
what's abstract and what's concrete...I'm content to leave it there because next we'll be discussing quantum interpretations...

...good discussions above regardless how things fall out...

 

[Dale]

I am certainly not going into a quantum discussion. GR is a classical theory.

I would like you to answer the first question about physical geometry. Do you agree that a table leg may be orthogonal (a geometrical relationship) to a table top in a physical and non-abstract sense?

If that geometrical relationship is non-abstract then calling other physical geometrical relationships "abstract" seems rather arbitrary.